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Physical Review A

Fisher entropy and uncertaintylike relationships in many-body systems

E. Romera, Juan Carlos Angulo, J. S. Dehesa

Published May 1, 1999• 34 citations
Publisher

Abstract

General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.

DOI
10.1103/physreva.59.4064

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General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.
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Rigorous and universal relationships among radial expectation values of any DDD-dimensional quantum-mechanical system are obtained, using R\'enyi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products $\ensuremath{\langle}{r}^{a}\ensuremath{\rangle}{}^{1/a}\ensuremath{\langle}{p}^{b}\ensuremath{\rangle}{}^{1/b}$, widely considered in the literature, including the famous Heisenberg relationship $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}\ensuremath{\geqslant}{D}^{2}/4$. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b>0a,b>0a,b>0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.
Rigorous bounds to atomic information entropies and their sum in position and momentum spaces have been derived in terms of radial and momentum expectation values ⟨ r n ⟩ and ⟨ p n ⟩. Numerical studies on atomic systems show that the bounds presented in terms of ⟨ r ⟩ and ⟨ p ⟩, the first moments of the position and momentum - space densities are sharper than those given by Gadre and Bendale in terms of ⟨ r 2 ⟩ and ⟨ p 2 ⟩, the second moments of the position and momentum-space densities. Within the BKCM procedure, several relationships between the information entropies and the average electron densities ⟨ρ⟩ and ⟨γ⟩ in position and momentum spaces have been established.